IR-TST 2011

In acute triangle $ABC$ angle $B$ is greater than $C$ . Let $M$ is midpoint of $BC$ . $D$ and $E$ are the feet of the altitude from $C$ and $B$ respectively. $K$ and $L$ are midpoint of $ME$ and $MD$ respectively. If $KL$ intersect the line through $A$ parallel to $BC$ in $T$ , prove that $TA=TM$ .

Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.

There are $n$ points on a circle ( $n>1$ ). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals $F$ such that for every element of $F$ like $A$ there is almost one other element of $F$ like $B$ such that $A \subseteq B$ (in this case we call $A$ is sub-interval of $B$ ). We call an interval maximal if it is not a sub-interval of any other interval. If $m$ is the number of maximal elements of $F$ and $a$ is number of non-maximal elements of $F,$ prove that $n\geq m+\frac a2.$

Define a finite set $A$ to be 'good' if it satisfies the following conditions:

• (a) For every three disjoint element of $A,$ like $a,b,c$ we have $\gcd(a,b,c)=1;$
• (b) For every two distinct $b,c\in A,$ there exists an $a\in A,$ distinct from $b,c$ such that $bc$ is divisible by $a.$

Find all good sets.

Find all surjective functions $f: \mathbb R \to \mathbb R$ such that for every $x,y\in \mathbb R,$ we have $$f(x+f(x)+2f(y))=f(2x)+f(2y).$$

The circle $\omega$ with center $O$ has given. From an arbitrary point $T$ outside of $\omega$ draw tangents $TB$ and $TC$ to it. $K$ and $H$ are on $TB$ and $TC$ respectively.a) $B'$ and $C'$ are the second intersection point of $OB$ and $OC$ with $\omega$ respectively. $K'$ and $H'$ are on angle bisectors of $\angle BCO$ and $\angle CBO$ respectively such that $KK' \bot BC$ and $HH'\bot BC$ . Prove that $K,H',B'$ are collinear if and only if $H,K',C'$ are collinear.b) Consider there exist two circle in $TBC$ such that they are tangent two each other at $J$ and both of them are tangent to $\omega$ .and one of them is tangent to $TB$ at $K$ and other one is tangent to $TC$ at $H$ . Prove that two quadrilateral $BKJI$ and $CHJI$ are cyclic ( $I$ is incenter of triangle $OBC$ ).

Find the locus of points $P$ in an equilateral triangle $ABC$ for which the square root of the distance of $P$ to one of the sides is equal to the sum of the square root of the distance of $P$ to the two other sides.

Let $p$ be a prime and $k$ a positive integer such that $k \le p$ . We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$ . (a) Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that $$f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),$$(b) Find a counter example for each prime $p$ and each $k > p$ .

We have $n$ points in the plane such that they are not all collinear. We call a line $\ell$ a 'good' line if we can divide those $n$ points in two sets $A,B$ such that the sum of the distances of all points in $A$ to $\ell$ is equal to the sum of the distances of all points in $B$ to $\ell$ . Prove that there exist infinitely many points in the plane such that for each of them we have at least $n+1$ good lines passing through them.

Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality $$\begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}$$ holds.

Let $ABC$ be a triangle and $A',B',C'$ be the midpoints of $BC,CA,AB$ respectively. Let $P$ and $P'$ be points in plane such that $PA=P'A',PB=P'B',PC=P'C'$ . Prove that all $PP'$ pass through a fixed point.